In each of the following, determine whether or not H is a...

In each of the following, determine whether or not $H$ is a subgroup of $G$. (Assume that the operation of $H$ is the same as that of $G$.) $ G=\langle\mathbb{R} \times \mathbb{R},+\rangle, H=\left\{(x, y): x^{2}+y^{2}>0\right\} . \quad H$ is $\square$ is not $\square$ a subgroup of $G$.

In each of the following, determine whether or not $H$ is a subgroup of $G$. (Assume that the operation of $H$ is the same as that of $G$.)
$ G=\langle\mathbb{R} \times \mathbb{R},+\rangle, H=\left\{(x, y): x^{2}+y^{2}>0\right\} . \quad H$ is $\square$ is not $\square$ a subgroup of $G$.

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Key Concepts

Subgroup

A subgroup is a subset of a group that itself forms a group under the same operation as the original group. This requires the subset to contain the identity element, be closed under the group operation, and include the inverses of its elements.

Closure Properties

Closure properties in the context of groups refer to the requirement that the set must be closed under the group operation as well as under taking inverses. This means that combining any two elements of the subset using the group operation, or taking the inverse of any element in the subset, must result in an element that still belongs to the subset.

Identity Element

The identity element in a group is a unique element that, when combined with any element of the group using the group operation, leaves the other element unchanged. For a subset to qualify as a subgroup, it must include this identity element.

Group

A group is a mathematical structure consisting of a set together with an operation that combines any two elements to form a third element, in a way that satisfies the axioms of associativity, the existence of an identity element, and the existence of inverse elements for every element in the set.

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